# L11a383

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a383 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 v^4-u^4 v^3-u^3 v^4+4 u^3 v^3-3 u^3 v^2-3 u^2 v^3+7 u^2 v^2-3 u^2 v-3 u v^2+4 u v-u-v+1}{u^2 v^2}$ (db) Jones polynomial $-4 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{2}{q^{7/2}}-9 q^{5/2}+\frac{4}{q^{5/2}}+10 q^{3/2}-\frac{7}{q^{3/2}}-q^{13/2}+2 q^{11/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-3} -6 z^5 a^{-3} -12 z^3 a^{-3} -8 z a^{-3} - a^{-3} z^{-1} +z^9 a^{-1} -a z^7+8 z^7 a^{-1} -6 a z^5+24 z^5 a^{-1} -12 a z^3+32 z^3 a^{-1} -8 a z+17 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -5 z^4 a^{-6} +z^2 a^{-6} +3 z^7 a^{-5} -8 z^5 a^{-5} +5 z^3 a^{-5} -z a^{-5} +4 z^8 a^{-4} +a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+22 z^4 a^{-4} +4 a^4 z^2-10 z^2 a^{-4} +3 z^9 a^{-3} +2 a^3 z^7-11 z^7 a^{-3} -7 a^3 z^5+18 z^5 a^{-3} +6 a^3 z^3-11 z^3 a^{-3} -a^3 z+7 z a^{-3} - a^{-3} z^{-1} +z^{10} a^{-2} +2 a^2 z^8+z^8 a^{-2} -5 a^2 z^6-14 z^6 a^{-2} +2 a^2 z^4+31 z^4 a^{-2} -a^2 z^2-16 z^2 a^{-2} + a^{-2} +2 a z^9+5 z^9 a^{-1} -7 a z^7-23 z^7 a^{-1} +15 a z^5+49 z^5 a^{-1} -22 a z^3-47 z^3 a^{-1} +9 a z+19 z a^{-1} - a^{-1} z^{-1} +z^{10}-z^8-3 z^6+10 z^4-10 z^2$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-5-4-3-2-10123456χ
14           11
12          1 -1
10         31 2
8        41  -3
6       53   2
4      65    -1
2     54     1
0    47      3
-2   34       -1
-4  14        3
-6 13         -2
-8 1          1
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.